I'm new to Lie theory and I'm trying to find all the rational elements of order 3 in the Lie group $A_2$. I found all the elements of order 3 contained in the fundamental region, which is defined by $w_1$ and $w_2$. Those points are the one listed right here :
$ \Big\{0, w_1, w_2, \frac{w_1}{3}, \frac{w_2}{3}, \frac{2w_1}{3}, \frac{2w_2}{3}, \frac{2w_1+w_2}{3}, \frac{w_1+2w_2}{3},\frac{w_1+w_2}{3}\Big\}. $
From what I understand, to find the rational elements of order 3, I have to take each of those points and multiply them by 3. Then, for each point, I consider the affine mirror, denoted $r_\xi$, defined by the line that joins $w_1$ and $w_2$, which is orthogonal to the highest root $\alpha_1 + \alpha_2$, and reflect those points. If the reflected point corresponds to its original coordinates, then it is a rational element of order 3. The thing is that I didn't find any rational element, but I found an article of A. Klymik and J. Patera about orbit functions which claims at page 50 that $\frac{w_1+w_2}{3}$ is a rational element. In fact, doing this method, I get that $\frac{w_1+w_2}{3}$ goes to the element $0$, so I don't understand what is my mistake. Any help would be appreciated.
Here's the arXiv link for the mentioned article.
Here's a picture of the fundamental region of $A_2$ and its fundamental weights and simple roots:
$A_2$" />